Learning from unordered sets is a fundamental learning setup, recently attracting increasing attention. Research in this area has focused on the case where elements of the set are represented by feature vectors, and far less emphasis has been given to the common case where set elements themselves adhere to their own symmetries. That case is relevant to numerous applications, from deblurring image bursts to multi-view 3D shape recognition and reconstruction. In this paper, we present a principled approach to learning sets of general symmetric elements. We first characterize the space of linear layers that are equivariant both to element reordering and to the inherent symmetries of elements, like translation in the case of images. We further show that networks that are composed of these layers, called Deep Sets for Symmetric Elements (DSS) layers, are universal approximators of both invariant and equivariant functions, and that these networks are strictly more expressive than Siamese networks. DSS layers are also straightforward to implement. Finally, we show that they improve over existing set-learning architectures in a series of experiments with images, graphs, and point-clouds.

翻译：从未排序的组合中学习是一个基本的学习设置,最近引起越来越多的注意。这一领域的研究侧重于一组元素由特性矢量代表的情况,而对于一组元素本身坚持其自身对称的常见情况则没有那么强调。该案例与许多应用有关,从分流图像爆发到多视图 3D 形状识别和重建。在本文中,我们提出了一个学习一般对称元素集的原则性方法。我们首先将线性层的空间描述为元素重新排序和元素内在对称的等空间,如图像的翻译。我们进一步显示,由这些层组成的网络,称为对称元素的深组,是变量和等离子函数的通用近似符,这些网络严格来说比Siamse网络更具有直观性。 DSS 层也比较简单。最后,我们显示,它们改进了以图像、图表和点焦格进行的一系列实验的现有定的学习结构。